Understanding Probability and Combinatorics: Key Concepts and Applications

1. What is Probability? • Quantification of uncertainty. • Mathematical model for things that occur randomly. • Random – not haphazard, don’t know what will happen on any one experiment, but has a long run order. • The concept of probability is necessary in work with physical biological or social mechanism that generate observation that can not be predicted with certainty. Example… • The relative frequency of such ransom events with which they occur in a long series of trails is often remarkably stable. Events possessing this property are called random or stochastic events week 1

2. Basic Combinatorics Multiplication Principle Suppose we are to make a series of decisions. Suppose there are c1 choices for decision 1 and for each of these there are c2 choices for decision 2 etc. Then the number of ways the series of decisions can be made is c1·c2·c3···. Example 1: Suppose I need to choose an outfit for tomorrow and I have 2 pairs of jeans to choose from, 3 shirts and 2 pairs of shoes that matches with this shirts. Then I have 2·3·2 = 12 different outfits. week 1

3. Example 2: The Cartesian product of sets A and B is the set of all pairs (a,b) where aA, bB. If A has 3 elements (a1,a2,a3) and B has 2 elements (b1,b2), then their Cartesian product has 6 members; that is AB = {(a1,b1), (a1,b2), (a2,b1), (a2,b2), (a3,b1), (a3,b2)}. • Some more exercise: 1. We toss R different die, what is the total number of possible outcome? 2. How many different digit numbers can be composed of the digits 1-7 ? 3. A questioneer consists of 5 questions: Gender (f / m), Religion (Christian, Muslim, Jewish, Hindu, others), living arrangement (residence, shared apartment, family), speak French (yes / no), marital status. In how many possible ways this questioneer can be answered? week 1

4. An order arrangement of n distinct objects is called a permutation. The number of ordered arrangements or permutation of n objects is n! = n · (n – 1) · (n – 2) · · ·1 (“n factorial”). By convention 0! = 1. The number of ordered arrangements or permutation of k subjects selected from n distinct objects is n · (n – 1) · (n – 2) · · · (n – k +1). It is also the number of ordered subsets of size k from a set of size n. Notation: Example: n = 3 and k = 2 The number of ordered arrangements of k subjects selected with replacement from n objects is n k. Permutation week 1

5. How many 3 letter words can be composed from the English Alphabet s.t: (i) No limitation (ii) The words has 3 different letters. 2. How many birthday parties can 10 people have during a year s.t.: (i) No limitation (ii) Each birthday is on a different day. 3.10 people are getting into an elevator in a building that has 20 floors. (i) In how many ways they can get off ? (ii) In how many ways they can get off such that each person gets off on a different floor ? 4.We need to arrange 4 math books, 3 physics books and one statistic book on a shelf. (i) How many possible arrangements exists to do so? (ii) What is the probability that all the math books will be together? Examples week 1

6. The number of subsets of size k from a set of size n when the order does not matter is denoted by or (“ n choose k”) . The number of unordered subsets of size k selected (without replacement) from n available objects is Important facts: Exercise: Prove the above. Combinations week 1

7. We need to select 5 committee members form a class of 70 students. (i) How many possible samples exists? (ii) How many possible samples exists if the committee members all have different rules? Example week 1

8. The Binomial Theorem • For any numbers a, b and any positive integer n • The terms are referred to as binomial coefficient . week 1

9. Multinomial Coefficients • The number of ways to partitioning n distinct objects into k distinct groups containing n1, n2,…,nk objects respectively, where each object appears in exactly one group and is • It is called the multinomial coefficients because they occur in the expansion Where the sum is taken over all ni = 0,1,...,n such that week 1

10. A small company gives bonuses to their employees at the end of the year. 15 employees are entitled to receive these bonuses of whom 7 employees will receive 100$ bonus, 3 will receive 1000$ bonus and the rest will receive 3000$ bonus. In how many possible ways these bonuses can be distributed? 2. We need to arrange 5 math books, 4 physics books and 2 statistic book on a shelf. (i) How many possible arrangements exists to do so? (ii) How many possible arrangements exists so that books of the same subjects will lie side by side? Examples week 1